
\prob{0025}{圆周角定理}

\begin{figure}[htbp]
  \centering
  \image{0025}
  \caption{0025：圆周角定理} \label{fig:0025}
\end{figure}

证明圆周角定理：一条弧所对圆周角等于它所对圆心角的一半。
\problabels{yellow/平面几何, green/证明题}

\subsection{等腰三角形} \label{subsec:0025-eqtri}

\begin{figure}[htbp]
  \centering
  \image{0025-eqtri}
  \caption{\nameref{subsec:0025-eqtri}：构造等腰三角形，然后利用角的关系证明。}
  \label{fig:0025-eqtri}
\end{figure}

基本思路：构造两个等腰三角形，通过一系列角的关系证明。

如图~\ref{fig:0025-eqtri}，连接$OA$。

\begin{align*}
  &\because   OA = OB = OC \\
  &\therefore \angle OAB = \angle OBA, \angle OAC = \angle OCA \\
  &\because   \angle OAB + \angle OBA + \angle AOB = 180^\circ \\
  &\therefore \angle AOB = 180^\circ - 2\angle OAB \\
  &\because   \angle OAC + \angle OCA + \angle AOC = 180^\circ \\
  &\therefore \angle AOC = 180^\circ - 2\angle OAC \\
  &\because   \angle AOB + \angle AOC + \angle BOC = 360^\circ \\
  &\therefore \angle BOC = 2(\angle OAB + \angle OAC) \\
  &\because   \angle OAB + \angle OAC = \angle BAC \\
  &\therefore \angle BOC = 2\angle BAC \\
\end{align*}

证毕。
